Arnold Conjecture (General Statement and Proof Outline)

Theorem8.1Arnold Conjecture

Let $(M^{2n}, \omega)$ be a closed symplectic manifold and $\phi \in \mathrm{Ham}(M, \omega)$ a Hamiltonian diffeomorphism with non-degenerate fixed points. Then $|\mathrm{Fix}(\phi)| \geq \sum_{k=0}^{2n} \dim H_k(M; \mathbb{Z}_2)$ .

The weak version (cuplength bound) states $|\mathrm{Fix}(\phi)| \geq \mathrm{cl}(M) + 1$ where $\mathrm{cl}(M)$ is the cuplength of $M$ .

Proof

The proof proceeds by Floer homology. Fixed points of $\phi = \phi_1^H$ are 1-periodic orbits of $X_H$ , which are generators of the Floer complex $CF_*(H)$ . The Floer differential $\partial$ counts rigid ( $\mathrm{ind} = 1$ ) Floer cylinders. Key steps:

$\partial^2 = 0$ : The boundary of the 1-dimensional moduli space of Floer cylinders consists of broken trajectories, giving algebraic cancellation.
Invariance: Continuation maps show $HF_*(H_0) \cong HF_*(H_1)$ for different Hamiltonians.
Computation: For $C^2$ -small $H$ (Morse function), Floer trajectories coincide with Morse gradient flow lines, so $HF_*(H) \cong HM_*(H) \cong H_*(M; \mathbb{Z}_2)$ .
Conclusion: $|\mathrm{Fix}(\phi)| = \mathrm{rank}(CF_*) \geq \mathrm{rank}(HF_*) = \sum_k \dim H_k(M; \mathbb{Z}_2)$ . $\square$

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RemarkTechnical Challenges

For general symplectic manifolds, the analysis requires: (a) compactification of moduli spaces using stable maps; (b) virtual perturbation techniques for transversality; (c) working over a Novikov ring to handle multiple homology classes. The full proof (Fukaya-Oh-Ohta-Ono, Liu-Tian, Pardon) uses Kuranishi structures or polyfold theory.